Approaches to Quantum Gravity

Three-dimensional spin foam Quantum Gravity 307The Lorentzian version of the Ponzano–Regge model is expressed in terms ofthe {6 j} symbols of the non-compact group SO(2, 1) [26]. Holonomies aroundparticles are SO(2, 1) group elements parametrized asg = P 4 + iκ P i τ i with P 2 4 + κ2 P i P i = 1, and P 4 ≥ 0, (16.55)with the metric (+ −−) and the su(1, 1) Pauli matrices, τ 0 = σ 0 ,τ 1,2 = iσ 1,2 .Massive particles correspond to the P i P i > 0 sector. They are described by ellipticgroup elements, P 4 = cos θ, κ|P| =sin θ. The deficit angle is given by the mass,θ = κm. All the mathematical relations of the Riemannian theory are translated tothe Lorentzian framework by performing the transformationP 0 → P 0 , P 1 → iP 1 , P 2 → iP 2 . (16.56)Note that this transformation differs from a usual Wick rotation (which rotatesP 0 only).The propagator remains given by the formula (16.41). The momentum spaceis now AdS 3 ∼ SO(2, 1). The addition of momenta is deformed accordinglyto the formula (16.31). We similarly introduce a group Fourier transform F :C(SO(2, 1)) → C κ (R 3 ) and a ⋆-product dual to the convolution product onSO(2, 1). Finally we derive the effective non-commutative field theory with thesame expression (16.43) as in the Riemannian case.It is also possible, in the context of Euclidean gravity, to take into account anon-zero cosmological constant . The corresponding model is the Turaev–Viromodel [29] based on U q (SU(2)), where q is on the unit circle for a positive cosmologicalconstant and q is real for a negative cosmological constant. For a positivecosmological constant, provides a maximal length scale. We wrote the explicitFeynman rules corresponding to this spin foam model in [4] and showed thatwe obtain a spherical or hyperboloid state sum based on the propagators on the3-sphere or the 3-hyperboloid respectively depending on the sign of . Furtherwork is needed to analyze the details of these models and extend the results to theLorentzian case.To sum up, we have shown how the Ponzano–Regge spin foam model canbe properly gauge fixed in order to provide a proper definition of 3d euclideanQuantum Gravity. We have seen how this model can be naturally coupled to matterand that the corresponding 3d Quantum Gravity amplitudes are actually theFeynman diagram evaluations of a braided and non-commutative QFT. This effectivefield theory describes the dynamics of the matter field after integration of thegravitational degrees of freedom. The theory is invariant under a κ-deformation ofthe Poincaré algebra, which acts non-trivially on many-particle states. This is anexplicit realization of a QFT in the framework of deformed special relativity (seee.g. [30]), which implements from first principles the original idea of Snyder [31]of using a curved momentum space to regularize the Feynman diagrams.